Finally, we leverage our deep learning algorithms for solving families of PDEs when varying the terminal condition.
In other words, the input is a function \(g_K\)  with parameter \(K\) , and the output is the solution to the PDE with terminal condition \(g_K\) .
This is performed by means of the class of DeepOnet neural networks aiming to approximate functional operators. These networks, introduced in [1]}, rely on a universal approximation theorem for operators [2]} stating that a neural network with a single hidden layer can approximate accurately any nonlinear continuous operator. The DeepOnet realizes this theorem in practice and can be used to learn the mapping between the terminal function of a PDE and its solution.
